Question

Jennifer and Boris are graduate assistants helping a professor grade student tests. Working together, they are able to complete the task in 1 hour and 12 minutes. If Jennifer could have graded the entire batch of tests by herself in 2 hours, how long would it have taken Boris to complete that task by himself?

Short answer

It would have taken Boris 3 hours to grade the tests by himself.

Step-by-step solution

Convert time to hours

First, convert 1 hour and 12 minutes into hours. There are 60 minutes in an hour. Therefore, 12 minutes is equal to 12/60 or 1/5 of an hour. Thus, 1 hour and 12 minutes is equal to 1 + 1/5 = 6/5 hours.

Set up the work rates

Let the rate at which Jennifer completes the work be J and the rate at which Boris completes the work be B. Since Jennifer can complete the task in 2 hours, her work rate is J = 1/2 (work per hour). Together, their combined work rate is 1/(6/5) = 5/6 (work per hour).

Express Boris's work rate

Given their combined work rate (J + B = 5/6) and Jennifer's work rate (J = 1/2), we can substitute Jennifer's work rate into the combined rate equation: 1/2 + B = 5/6.

Solve for Boris's work rate

Subtract Jennifer's work rate from the combined rate to find Boris's work rate: B = 5/6 - 1/2. To subtract these fractions, find a common denominator: 5/6 - 3/6 = 2/6 = 1/3.

Find the time it takes Boris

Since Boris's work rate (B) is 1/3 work per hour, it will take him a time equal to the reciprocal of his work rate to complete the task alone: Time = 1/B = 1/(1/3) = 3 hours.

Key concepts

Collaborative Work Problem

A collaborative work problem involves multiple individuals working together to complete a task.

In this particular problem, Jennifer and Boris are grading tests. Knowing how long it takes them to work together can help us determine how long it would take each of them to do the work individually.

Understanding collaborative work problems can help whether you are tackling homework or planning projects in real life.

It's essential to grasp how combining work rates for different people or machines can lead to quicker completion times.

Work Rates

Work rates describe the speed at which someone or something completes work. In our exercise, Jennifer's work rate is 1 test batch per 2 hours, or \( J = \frac{1}{2} \) batches per hour.

The combined work rate of Jennifer and Boris working together is calculated as \( \frac{5}{6} \) batches per hour, because they complete the task in 1 hour and 12 minutes.

To find individual work rates, we set up and solve equations where their rates are added together, helping us to understand how they contribute to the shared workload.

This foundational concept is essential for problems relating to efficiency and team productivity.

Fractions in Time Calculations

Fractions are crucial in time calculations for problems like this one. We first converted 1 hour and 12 minutes into hours: \( 1 + \frac{12}{60} = \frac{6}{5} \) hours.

We also used fractions to work with rates. For example, Jennifer's rate is \( \frac{1}{2} \) batches per hour.

Subtraction and addition of fractions were needed to isolate Boris's work rate. Keeping track of numerators and denominators, and understanding common denominators, are necessary skills for solving these problems.

Don't shy away from practicing these calculations, as they form the backbone of solving collaborative work questions.

Reciprocal Work Rate

The reciprocal work rate comes into play when we want to find the total time taken for a task based on the rate of work done. It is essentially the inverse of the work rate.

In Jennifer and Boris's problem, Boris's work rate was found to be \( \frac{1}{3} \) batches per hour. To determine the time it would take him alone, compute the reciprocal: \( \frac{1}{ \frac{1}{3} } = 3 \) hours.

This step is critical. It converts the rate of completion into the actual time, solving the problem. Understanding this helps in breaking down how long tasks will take when individual work rates are known.